This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain’s geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.

Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool.